M.Sc. Student of Geophysics, Earth Physics Department, Institute of Geophysics, University of Tehran, Iran
Assistant Professor, Earth Physics Department, Institute of Geophysics, University of Tehran, Iran
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The ill-posed inverse problems play an important role in various fields of geophysical studies. Basic information about geophysical models (the unknown parameters) is needed to find a unique and stable solution to such problems. Recent progresses in computational methods and advances in analysis of real world signals provide us suitable tools for extracting more detailed information about geophysical models from noisy, uncertain observations (recorded data). In this paper, we study relative travel time estimation of the individual seismic arrival times using total variation (TV) regularization, a regularization method which has recently attracted much attention of scientist for reconstruction of models having sharp boundaries.
Seismic waves convert to different phases when passing through the boundary of earth layers having different geological properties and structure of the materials though which the waves propagate. For example, seismic travel times can be used to inversely determine the velocity field of the area under study via tomography. Furthermore, boundary layer structure at the core-mantle boundary (CMB) can be investigated using SKS and SPdKS timing difference that are recorded by a broadband seismometer arrays. Therefore, accurate measurement of the travel time of seismic phases or their differences is very important. However, robust measurement of the travel time is often difficult, specifically when data are contaminated by noise and lacks clearly defined onsets. Travel time of a particular phase can be determined by several methods including cross-correlation technique and hand picking. The former is done by cross-correlation between the signal of interest and a reference phase. Determination of the reference phase is a major challenge as the accuracy of the process depends significantly on the similarity of it to the desired phase which is to be studied. Hand picking is also challenging because background noise often obscures confident identification of signal initiation. Furthermore, seismic phases are often altered due to scattering, attenuation, multipathing, or anisotropy, making accurate measurements of their travel times even more difficult. Considering these, a preprocessing of the signal is required to improve the signal to noise ratio and sharpen signal onsets such that the process of determining arrival time is more robust.
In this paper, we address the problem of accurate determination of seismic arrival times or relative times of different phases. We formulate the effects of source, attenuation, and receiver structure by convolution with a skewed Gaussian and try to remove those effects from the seismogram via deconvolution. Deconvolution is a longstanding problem in many areas of signal and image processing with applications in astronomy, remote-sensing imagery, medical imaging, and other fields working with imaging devices.
Two possible deconvolution scenarios are nonblind, where the Gaussian function, considered for degradation, is assumed to be known a priori, and blind, where the Gaussian function is not known a priori. Even in the presence of the perfect degradation, restoration of the path from the observed seismogram is an ill-posed problem needing an appropriate prior to make the solution unique and stable. Numerous algorithms have been developed to address the problem; including least squares type Wiener filter and more sophisticated regularization methods like total variation (TV) regularization. Although the methods can provide satisfactory results, they are generally nonblind and therefore require a good knowledge of the Gaussian blurring function in order to work properly. In reality, however, the degradation function is not known with good accuracy. Therefore, it should also be estimated during deconvolution making the problem more ill-posed.
The blind deconvolution which is used in this paper uses a sequential approach where the Gaussian function is first estimated from the data via an L-curve analysis. The estimation of first step is later used in combination with TV yields a piece wise constant reconstruction and preserves the edges of the signal that is important for defining onsets and so accurate measurement of the travel time of seismic phases. TV helps to stabilize the deconvolution and at the same time preserve the discontinuities of the solution. It improves the signal to noise ratio, sharpens seismic arrival onset, and acts as an empirical source deconvolution, thus enables more reliable relative travel time estimation of phase initiation.
Instead of the conventional l1-norm used for TV functional, here, we use a more sparsifying potential function for the purpose of sharpening the phase onsets. Due to simplicity and good convergence, an Iteratively Reweighted Least Squares (IRLS) method is used for optimizing the generated objective function.
Two different examples are used to investigate the performance of the proposed algorithm: (1) signal restoration of SKS and S (or Sdiff) in synthetic seismograms and (2) the restoration of actual data for 30 seismic recording of a deep focus south American earthquake. The obtained results confirm high performance of the proposed method in calculating time difference of these phases.